There is the possibility Chelsea may be ready to look up in the Premier League and it is a line Mauricio Pochettino is keen to push. This was a third win on the spin in the competition, a reaction to the 1-0 midweek loss at Middlesbrough in the Carabao Cup semi-final first leg and all of the criticism that followed.

Yet Pochettino has to know his team continues to look callow, a little fragile; short of cohesion and punch in the areas that matter the most. When they win at the moment, they are doing just enough, putting their supporters through the mill and leaving them with many questions. It was the case again here, with Chelsea grateful to Cole Palmer, their star of an erratic season, and his prowess from the penalty spot.

Related: Fulham’s Marco Silva says VAR should have called for Malo Gusto red card

Palmer provided the only sparks of a forgettable game that finished in a fifth straight away league defeat for Fulham, his skills and close control, his eye for a pass so easy on the eye. When opportunity knocked for him after Issa Diop had fouled Raheem Sterling just before half-time, he duly converted his fifth penalty out of five since his move from Manchester City last summer. He has nine goals for his new club, plus seven assists.

Fulham had plenty of reference points for their poor away form. Only two clubs in the division have a worse record on their travels while they had not won here since 1979. They have not scored a goal here since 2011.

Marco Silva was furious the game was not shaped differently by a red card on 38 minutes for the Chelsea right-back, Malo Gusto, who stretched into a tackle on Willian and got nowhere near the ball.

Gusto clattered his studs into the former Chelsea winger’s shin and after the referee, Anthony Taylor, showed a yellow card, it felt like one for the video assistant referee to refer for an upgrade. Instead, Taylor’s decision was allowed to stand. Gusto, who was dismissed for something similar against Aston Villa in September, got away with it. Across the piece, though, Fulham did not do enough to influence what they could control.

It was subdued in the home seats at the outset, most of the noise coming from the travelling support, and it seemed to chime with the general sense of bemusement with this Chelsea team. Pochettino was agitated until the penalty, gesturing to his assistants in obvious irritation, howling in anger at one point.

Chelsea did not deserve to be in front at the interval but they were after Palmer provided one of the precious few moments of incision. He pinged a nice pass up the inside-right channel for Sterling and it was a clumsy tackle from Diop after the winger had jinked inside. Sterling felt the contact on his trailing leg and moments later Palmer could perform his shivering goal celebration after a cold-blooded conversion.

Chelsea squeezed high. They controlled the first half but there was not much from them in the attacking third. The Fulham fans roared as their players flew into challenges and it was all too plodding and predictable from Chelsea; too much back and sideways stuff.

Enzo Fernández had an early shot blocked and Conor Gallagher shot high after a quick Palmer pass but Chelsea’s only clear chance before the penalty came when Fernández crossed and Armando Broja headed wide.

Fulham were no better in an attacking sense; far too meek. It said everything when Axel Disasi left a cross alone in the 44th minute, sparking a moment of panic in the Chelsea six-yard box, that there was no Fulham player nearby to capitalise. Andreas Pereira drew a routine save out of Djordjie Petrovic and Harry Wilson a much better one at the near post after a dangerous Antonee Robinson cross.

The Carabao Cup had framed the game, Chelsea feeling the burn from Middlesbrough, Fulham much more positive after their . The prospect of a first west London derby cup final remains alive.

Palmer aside, Chelsea continued to make hard work of things in the ­second half. Their fans groaned when Broja wanted too many touches or to check back; they needed something more direct.

Sterling headed against a post on 53 minutes although he looked offside and Levi Colwill would lift wastefully high. Could Fulham fashion a late sting? The big chance was the one that fell to Raúl Jiménez in the 73th minute after Pereira had crossed low and Fernández slipped but he shot too close to Petrovic. There was nervousness for Chelsea at the end as Fulham loaded men forward, with Willian also blowing a free-kick on the edge of the area.

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It was Chelsea who went closer to scoring. Gallagher hit the outside of the post with the outside of his right boot and the substitute Noni Madueke extended Bernd Leno.

There had been cheers from the Chelsea crowd when Ben Chilwell got on for his first football since late September as a 77th-minute substitute. The overriding emotion for them was relief.